Photonic Quantum-to-Quantum Bernoulli facotry

Randomness generation and manipulation algorithms are a cornerstone of information theory and in cryptography. In this context, the inherently random nature of quantum mechanics has proved to be a useful resource, leading to the realisation of several quantum information protocols. Within this framework, there is an increasing search for new techniques for generating and manipulating quantum randomness that can be used as sub-routines for quantum protocols. In recent years, there have been various proposals for quantum generalisations of a family of classical algorithms known as Bernoulli factories [1] which, starting from extractions of a Bernoulli distribution with unknown parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$p$</tex>, allow one to simulate exactly a new Bernoulli distribution with parameter <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f(p)$</tex> for a given function <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$f(x):\mathscr{D}\subseteq[0,1]\rightarrow[0,1]$</tex>. In particular, it was shown that the use of quantum resources results in an advantage in terms of implementable functions [2], [3] and the average number of samples required by the algorithm.